My way of approaching this would be, what's the worst thing that could happen?
First: if the car is having any safety-related issues - brakes, tires, steering, wipers -- don't do it. Rent a car. You particularly don't want to have one of these items fail while on the road.
If the tires, brakes, and steering are good -- essentially, the car is safe to drive -- then the worst that could happen is that the car breaks down partway through the trip. You'd need either roadside service or a ride, so is cell coverage OK throughout your route? If it is, do you have AAA or the like to get the tow/ride? If you might not have cell coverage, I'd recommend renting again. If you will for sure, make sure your AAA membership is up to date and go ahead. You might end up having to have the car towed further than AAA will cover; you'll need to decide if that's worth it or not. If it's not, rent the car.